No decisions have been made about how or where to replace the 66-year-old Port Authority Bus Terminal in Manhattan, which is overrun by over 115,000 daily commuters from New Jersey and beyond.

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After three years of planning for a new bus terminal in Manhattan, the commissioners of the Port Authority of New York and New Jersey still agree that New York City badly needs one. But, as the nasty debate over paying for it has spilled out in public, they appear to agree on little else.

The discord has threatened to paralyze the board that oversees the agency and that is responsible for transportation projects critical to the region. When the time came on Friday to publish the board’s monthly agenda, the agency punted, hinting that Gov. Andrew M. Cuomo of New York, a Democrat, might instruct his appointees not to conduct any business, just as he did before the board’s previous meeting.

When the agenda for the meeting this week finally appeared on Monday afternoon, it suggested that politicians from the opposite sides of the Hudson River had still not bridged their differences. They may still be billions of dollars apart in their views on how the agency should spend the money it collects from tolls and transportation fees.

The battle over the bus terminal shows just how quickly the Port Authority can fall into the kind of dysfunction that allowed appointees of Gov. Chris Christie of New Jersey, a Republican, to spitefully close lanes at the George Washington Bridge in 2013. Mr. Christie has since stepped back from the agency, but the tussle for influence over its vast finances has remained fierce.

It has caused at least one of Mr. Cuomo’s associates to quit abruptly and has drawn in city and state officials from both states, as well as Representative Jerrold Nadler, a Manhattan Democrat. Accusations of bad-faith bargaining have flown back and forth. Mr. Nadler went so far as to question the motivations of the agency’s chairman, John J. Degnan, and to call on him to recuse himself from deliberations over the bus terminal.

On Monday, Mr. Nadler declined to discuss his allegation. But he said of Mr. Degnan, “All I do know is that the New York people at the Port Authority, and the governor’s people, they all say that he’s been just dictatorial on everything.”

Angered by the attacks, Mr. Degnan, a Christie appointee, has planted his feet and squared up for a fight. “I am the chairman of the Port Authority,” he said. “The Port Authority needs to work with the governors, but it should not be submissive to either one of them.”

No decisions have been made about how or where to replace the 66-year-old terminal, a destination widely derided as forlorn, and one that is overrun by over 115,000 daily commuters from New Jersey and beyond. But there is general agreement that the solution must include a new or revamped terminal within a few blocks of the existing one, on Eighth Avenue in Manhattan.

The crux of the current dispute is how much money the authority should commit to the project. The commissioners have been wrestling all year with a revision of the agency’s 10-year capital plan. That budget, drawn up in 2014, did not include any money for the bus terminal.

Facing cost estimates of $3.7 billion to $15.3 billion, Mr. Degnan has demanded that at least $3.5 billion of the capital plan be designated for the terminal. But on Tuesday, Richard Azzopardi, a spokesman for Mr. Cuomo, said the governor would agree to that only if at least two-thirds of the total was effectively contributed by New Jersey. “Asking New Yorkers to shoulder the burden of a $10 billion project many of them won’t ever use is a bad deal and a non-starter,” he said.

Jameson W. Doig, a professor emeritus at Princeton who has chronicled the history of the Port Authority, said he had been told that Mr. Cuomo did not want his appointees to attend the board meeting in November while the dispute was unresolved. One of them, Steve Cohen, who was the vice chairman of the board, skipped the meeting and resigned.

Mr. Cohen was the second Cuomo appointees to depart in the last few months, leaving the New York commissioners outnumbered, six to four. The previous vice chairman, Scott Rechler, left this fall.

Mr. Doig said although there had been gridlock at the agency in the past, he could not recall another instance of a governor’s asking commissioners to boycott a board meeting.

“We know that Governor Cuomo has been very interested in having substantial funds to carry out projects that he thinks are important,” he said, alluding to Mr. Cuomo’s championing of an overhaul of La Guardia Airport. “From his point of view, the bus terminal is not one of them.”

Mr. Azzopardi said: “The law provides that either governor can veto any action of the Port Authority. So if someone wants to run the Port Authority, they should run for governor.”

Mr. Nadler said he thought $2 billion was fair because the Port Authority was including the same amount in the capital plan for a project to build train tunnels under the Hudson River, known as Gateway, which is critical to improving travel in the region.

No draft of the revised capital plan has been released, and the agency has not said how much would go for the Gateway project. Steve Coleman, a spokesman for the agency, said late on Monday that he had no answers to questions about the capital plan.

It was not clear whether the commissioners would have answers when they gather for their monthly board meeting on Thursday. The delayed agenda indicated that they would vote on a resolution to publish a draft of the capital plan by Dec. 19.

After that, the resolution says, the agency would invite the public to join in the debate over how it should spend more than $28 billion in capital funds. Public hearings will be held before any final decisions are made, according to the resolution.

That is a very different process than the authority has used in the past, though it is, essentially, the same process the agency adopted to ram through an unpopular, steep toll increase in 2011.

The public hearings are likely to draw crowds, given the strong reactions to the agency’s previous decisions about the bus terminal.

New Jersey officials fear the agency will decide to build a satellite depot in their state, forcing many bus riders to transfer to PATH trains to reach Manhattan. Residents of the Hell’s Kitchen neighborhood, which surrounds the terminal, fear eminent domain will be invoked to take private property for the project.

Last week, a group of New Jersey officials gathered in Hackensack, N.J., to call for a commitment to fully fund a new terminal on the West Side of Manhattan that would preserve a one-seat ride for commuters. Stephen M. Sweeney, a New Jersey Democrat who is the president of the State Senate, said $2 billion would be too little to ensure that the project moved forward.

State Senator Loretta Weinberg, a Bergen County Democrat, said $2 billion was “ not nearly enough to guarantee a commitment to the bus terminal.” Ms. Weinberg said she thought that she and the New Jersey officials had reached the framework of an agreement more than a month ago, and was caught off-guard when the debate turned hostile. Ms. Weinberg said she hoped “cooler heads will prevail” when the commissioners meet and try to wrap up the capital plan.

Mr. Doig said he believed that the infighting was avoidable.

“The kind of severe political meddling that we’ve seen in the last six years is not inescapable,” he said. “You need to have governors who think about the important issues the region faces as opposed to thinking, Let’s find a way to use the Port Authority and its money to enhance my reputation.”